A mathematical model for polymer lens shrinkage

Anteryon BV produces optical lenses at the wafer scale, which have applications in several areas, the predominant one being in mobile phone cameras. The basic components of a wafer lens are a polymer, which satisfies the optical requirements, and a glass substrate on which this is bonded. During the manufacturing process, the polymer shrinks due to the polymerisation and as a result the final product will not have the designed shape. We have succeeded in modelling this deformation of the rotationally symmetric lens, using the principles of linear elasticity, both for an incompressible and for a more realistic, compressible medium. We have formulated the governing differential equations for the system, under a set of assumptions and used a perturbation technique based on the smallness of the aspect ratio e, to obtain analytical solutions which include the O(e^2) terms, for the displacements of the material points of the lens. These displacements are used to formulate an analytical formula for the resulting lens shape, after a certain amount of shrinkage, given by the parameter zeta has taken place. With the second order terms, the developed model is an improvement on the existing models at Anteryon, which correspond with the zeroth order terms of our model. Using this analytical formula, we also studied and developed analytical formulae for the inverse problem, i.e., what should be the mould shape in order to achieve the target lens shape. We have illustrated the developed model by applying it to a theoretical spherical mould. We have validated the model by comparing it to the measurements of moulds and lenses provided by Anteryon. We have applied the model to measured mould-lens combinations for many positions on the wafer, and for different lens shapes. For a given wafer, we have optimised the parameter zeta by considering the Root Mean Square value of the difference between the model output and the observed lens shape for each combination. The main results are a validation of our model and a method to determine the shrinkage parameter zeta, under specified process conditions. With this value of zeta, and the inverse model, we have a method to predict the ideal mould shape for a target lens shape. In the most ideal case, only one iteration step would be needed to predict this mould shape. Finally, we have also proposed a relation to determine Poisson's ratio for the medium; this result is based on a comparison between the incompressible and compressible models and the measurements.